The Public ReST API is based on the web script framework, but with significant abstractions / enhancements built on top. If you develop custom web scripts in Alfresco, you are using just the bare bones web script framework and add endpoints under a separate base URL.

Both web scripts and ReST API can be called by any ReST client, of course assuming that the web scripts you want to call are actually implemented as ReST-ful services (web scripts does not require any compliance to ReST).

You can also run API Explorer against your local Alfresco system. Simply install the API Explorer webapp on your server and you should be good to go. The project's sources can be found on GitHub and the WAR is available from Alfresco Nexus.